Ch02 Rigid Bodies in Plane and 3-D Motion.docx

KINEMATICS OF RIGID BODIES IN PLANE AND 3-D MOTION 15.90 (Beer & Johnston) 368309906000 The disk shown has a constant angular velocity of 500 r/min counterclockwise. Keeping that rod BD is 250 mm long, determine the acceleration of collar D when (a) = 90, (b) = 180. (a) = 90 Velocity Analysis B rotates about a fixed axis through A D translates BD is in general plane motion (B is chosen as reference point) i – components: 0.229 BD = 0 BD = 0 Acceleration Analysis 5588008445500 167640013843000 i-components: 137.08 + 0.229BD = 0 BD = = -598.6 rad/s2 j-components: aD = 0.1BD = -0.1 (-598.6) = 59.9 m/s2 Vector Polygon 0838200.2299BDi 0.1BDj aD/Bt aB aD 000.2299BDi 0.1BDj aD/Bt aB aD Not to scale! b) = 180 Velocity Analysis i-components: 2.62 + 0.2 BD = 0 BD = = -13.1 rad/s Acceleration Analysis 5937257620000 i-components: 0 = 0.2BD + 25.74 BD = = -128.7 rad/s2 j-components: aD = 137.08 – 0.15BD + 34.32 = 190.7 m/s2 Vector Polygon 329565160020aD 0.15BDj aD/Bt 25.74 i aD/Bn aB 35.32i 0.2BDi 00aD 0.15BDj aD/Bt 25.74 i aD/Bn aB 35.32i 0.2BDi Not to scale! 153543017843500 Example: In the four-bar linkage shown, control link OA has a counterclockwise angular velocity 0 = 10 rad/s during a short interval of motion. When link CB passes the vertical position shown, point A has coordinates x = -60 mm and y = 80 mm. Determine, by means of vector algebra, the angular velocity of AB and BC. Link AO is in rotation about a fixed axis through 0 Link CB is in rotation about a fixed axis through C Link AB is in general plane motion j-components: 0 = -600 + 240AB AB = 600/240 = 2.5 rad/s i-components: -180WBC = -800 - 100AB 180BC = 800 + 100(2.5) BC = 1050/180 = 5.83 rod/s 15.93 271462534290A B j i 0.15m 0.075m aC/An aC/At aB/An VB/At aB/At 00A B j i 0.15m 0.075m aC/An aC/At aB/An VB/At aB/At 0000 AB rotates with a constant angular velocity of 60 r/min clockwise. Knowing that gear A does not rotate, determine the acceleration of the tooth of gear B which is in contact with gear A. Velocity Analysis B rotates about a fixed axis through A Gear A does not rotate C is the instantaneous center of rotation of gear B Acceleration Analysis 52387514795500 8032759969500 Note: Gear B is in general plane motion; B is chosen as reference point. Vector Polygon 27432048260aC/Bn aC aB/An 00aC/Bn aC aB/An Not to scale! RATE OF CHANGE OF A VECTOR WITH RESPECT TO A ROTATINT FRAME OF REFERENCE 321945124460IA/B IB IA X X1 Y1 B A Y 00IA/B IB IA X X1 Y1 B A Y XY frame is fixed xy frame rotates with angular velocity about he z-axis (i.e. perpendicular to plane of screen) not fixed since xy rotating. Evaluation of and 101155562865di = (1) d d d di = j d j i dj/dt = -id/dt di/dt = jd/dt 00di = (1) d d d di = j d j i dj/dt = -id/dt di/dt = jd/dt Introduce cross-product 187642543815i k j 00i k j Generalization For any vector A 1188720377825X11 |dA|XY Ad Y1 Y X1 X |dA|x1y1 dA=d(|A|) Y11 Ad dA d 00X11 |dA|XY Ad Y1 Y X1 X |dA|x1y1 dA=d(|A|) Y11 Ad dA d Background Vector A swings to A1 in time dt observer attached to frame xy (i.e. rotating frame) sees that consists of two components. - A dB/dt due to rotation of A through d/B in xy. - dA/dt due to change in magnitude of A. Part of absolute rate of change is A not seen by rotating observer is . A is magnitude of vector A. Plan motion in a rotating frame Acceleration 3905885-11620500 3145155-3048000 normal or centripetal acceleration due to rotation of rotating frame tangential acceleration due to angular acceleration of rotating frame 3333753810X x1 y1 Y A B 00X x1 y1 Y A B 2VAB – CORIOLIS ACCELERATION 83820010795000 5238758509000 174625014160500 66357515494000 6635751397000101282513970003143251397000 1828803048000 Consider a rotating disk with a radial slot A small particle A is confined to slide in the slot Let = constant and Vrel = constant The velocity of A has two components: x (due to rotation of the disk) vrel (due to motion of A in the slot) 90678019050000 Consider the rate of change of the velocity of A: - no change in magnitude of Vrel since Vrel = constant. - change in direction of Vrel is - change in magnitude of x is dx - change in direction of x is xd Rates of change are: are in the (+) y-direction is in the (-) x-direction Total rate of change of VA: (normal) (Coriolis) since Vrel = constant and slot has no curvature since is constant 365760-9144000 XY : Fixed Frame xy : Rotating Frame Recall for a fixed frame: Now for a rotating frame: 017526000 XY : Fixed Frame Xy : Rotating Frame : normal acceleration of a point (P) fixed in the rotating frame : tangential acceleration of a point (P) fixed in the rotating frame : acceleration of point A in the rotating frame : Coriolis acceleration brought about by the rotating () of the rotating frame and relative motion (Vrel) in the rotating frame 3840480-914400015.119 The motion of pin P is guided by slots cut in rods AE and BD. Knowing that the rods rotate with the constant angular velocity A = 4 rad/s ? and B = 5 rad/s ?, determine the velocity of pin P for the position shown. Pin P moves in BD and AE both of which rotate relative motion in a rotating frame 7683502095500 8032757810500 Equateand Coordinate transformation: 7467601016030o 30o i* j j* i 0030o 30o i* j j* i j-component: -1.1547 + 0.722 sin 30 + VP/BD cos 30 = 0 i-components: -VP/AE + 0.722 cos 30 - VP/BD sin 30 = 0 2286000147955Vp -0.167i -1.155j 00Vp -0.167i -1.155j or 229171526035-0.722i* Vp -0.916j* 00-0.722i* Vp -0.916j* 3619500-1828800015.123 At the instant shown the length of the boom is being decreased at the constant rate of 150 mm/s and the boom is being lowered at the constant rate of 0.075 rad/s. Knowing that = 30, determine (a) the velocity, (b) the acceleration of point B. There is relative motion of B in the rotating x-y frame 59372512319000 3627120152400-0.15i -0.45j VB 00-0.15i -0.45j VB (a) 2863850116205008731251162050052387511620500 (b) 3811905156210aB 0.023j -0.034i 00aB 0.023j -0.034i 356616011938000 The vertical shaft and attached clevis rotate about the z-axis at the constant rate  = 4 rad/s. Simultaneously, the shaft B revolves about its axis OA at the constant rate 0 = 3 rad/s, and the angle is decreasing at the constant rate of /4 rad/s. Determine the angular velocity and the magnitude of the angular acceleration of shaft B when = 30. The x-y-z axes are attached to the clevis and rotate with it. 366712527305005867400273050026543005511800017462550355500136207527305007334252730500 1463040000 3749040-28575001. The circular plate and rod are rigidly connected and rotate about the ball-and-socket joint ( ) with an angular velocity = i + j + k. Knowing that VA = -(540 mm/s)i + 350 mm/s)j + (r4)2k and ij = 4 rad/s. Determine (a) the angular velocity of the assembly, (b) the velocity of point B. 018351500 2. A disk of radius r rotates at a constant rate 2 with respect to the are ( ), which itself rotates at a constant rate 1 about the Y axis. Determine (a) the angular velocity and angular acceleration of the disk, (b) the velocity and acceleration of point A on the rim of the disk. 4147820-124460003. The bent rod ABC rotates at a constant rate 1. Knowing that the collar D moves downward along the rod at a constant relative speed u, determine for the position shown (a) the velocity of D, (b) the acceleration of D. 7239017526000 4. A disk of radius r spins at the constant rate 2 about an axle held by a fork-ended horizontal rod which rotates at the constant rate 1. Determine the acceleration of point I for an arbitrary value of the angle .